Special divisor

In mathematics, in the theory of algebraic curves, certain divisor on an algebraic curve on a curve C are particular, in the sense of determining more compatible functions than would be predicted. These are the special divisors. In classical language, they move on the curve in a larger linear system of divisors.

The condition to be a special divisor D can be formulated in sheaf cohomology terms, as the vanishing of the H 1 cohomology of the sheaf of the sections of the invertible sheaf or line bundle associated to D . This means that, by the Riemann-Roch theorem, the H 0 cohomology or space of holomorphic sections is as large as possible (there is the minimum obstruction to taking a section).

Alternatively, by Serre duality, the condition is that no holomorphic differential with divisor ≥ − D exists on the curve.

=Brill-Noether theory=

The Brill-Noether theory in algebraic geometry is the theory of special divisors on generic algebraic curves. It is of interest mainly in the case of genus (mathematics)

: g ≥ 3.

In conceptual terms, for g given, the ) of a given degree d , as a function of g , that must be present on a curve of that genus.

The theory is named for the German geometers Ludwig Brill and Max Noether. The results were given in nineteenth century style; the whole theory was updated and modern proofs given by Phillip Griffiths and others.

These formulations can be carried over into higher dimensions, and there is now a corresponding Brill-Noether theory for some classes of algebraic surfaces.